Observation of $D^{+}\to K^{+}{\eta }^{(}{\prime }^{)}$ and Search for $CP$ Violation in $D^{+}\to {\pi }^{+}{\eta }^{(}{\prime }^{)}$ Decays

We report the first observation of the doubly Cabibbo-suppressed decays $D^{+}\to K^{+}{\eta }^{(\prime )}$ using a $791{\mathrm{fb}}^{-1}$ data sample collected with the Belle detector at the KEKB asymmetric-energy $e^{+}{e}^{-}$ collider. The ratio of the branching fractions of doubly Cabibbo-suppressed relative to singly Cabibbo-suppressed $D^{+}\to {\pi }^{+}{\eta }^{(\prime )}$ decays are $\mathcal{B}(D^{+}\to K^{+}\eta )/\mathcal{B}(D^{+}\to {\pi }^{+}\eta )=(3.06\pm 0.43\pm 0.14)\%$ and $\mathcal{B}(D^{+}\to K^{+}{\eta }^{\prime })/\mathcal{B}(D^{+}\to {\pi }^{+}{\eta }^{\prime })=(3.77\pm 0.39\pm 0.10)\%$. From these, we find that the relative final-state phase difference between the tree and annihilation amplitudes in $D^{+}$ decays, ${\delta }_{\mathrm{TA}}$, is $(72\pm 9)°$ or $(288\pm 9)°$. We also report the most precise measurements of $CP$ asymmetries to date: $A_{CP}^{D^{+}\to {\pi }^{+}\eta }=(+1.74\pm 1.13\pm 0.19)\%$ and $A_{CP}^{D^{+}\to {\pi }^{+}{\eta }^{\prime }}=(-0.12\pm 1.12\pm 0.17)\%$.

Figure 1

The invariant mass distributions of $h^{+}\eta$ ($h^{+}{\eta }^{\prime }$) in the left (right) plot after the initial selection. The solid histograms show ${\pi }^{+}{\eta }^{(\prime )}$ while the dashed histograms show $K^{+}{\eta }^{(\prime )}$ final states. The two inset histograms are $K^{+}{\eta }^{(\prime )}$ decays with enlarged vertical scales.

Figure 2

The invariant mass distributions used for the branching fraction measurements. The top two plots are for the ${\pi }^{+}\eta$ (left) and ${\pi }^{+}{\eta }^{\prime }$ (right) final states while the bottom two plots are for the $K^{+}\eta$ (left) and $K^{+}{\eta }^{\prime }$ (right) final states. Points with error bars and histograms correspond to the data and the fit, respectively.